cp's OEIS Frontend

A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.

%I A363488 #8 Jul 08 2023 23:06:34
%S A363488 2,4,6,8,10,12,14,16,20,22,24,26,28,32,34,36,38,40,44,46,48,52,56,58,
%T A363488 60,62,64,68,72,74,76,80,82,84,86,88,92,94,96,100,104,106,112,116,118,
%U A363488 120,122,124,128,132,134,136,140,142,144,146,148,152,156,158,160
%N A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.
%C A363488 The multiset of prime factors of n is row n of A027746.
%C A363488 Also numbers whose prime factors have low median 2, where the low median (see A124943) is either the middle part (for odd length), or the least of the two middle parts (for even length).
%e A363488 The terms together with their prime indices begin:
%e A363488      2: {1}            34: {1,7}             72: {1,1,1,2,2}
%e A363488      4: {1,1}          36: {1,1,2,2}         74: {1,12}
%e A363488      6: {1,2}          38: {1,8}             76: {1,1,8}
%e A363488      8: {1,1,1}        40: {1,1,1,3}         80: {1,1,1,1,3}
%e A363488     10: {1,3}          44: {1,1,5}           82: {1,13}
%e A363488     12: {1,1,2}        46: {1,9}             84: {1,1,2,4}
%e A363488     14: {1,4}          48: {1,1,1,1,2}       86: {1,14}
%e A363488     16: {1,1,1,1}      52: {1,1,6}           88: {1,1,1,5}
%e A363488     20: {1,1,3}        56: {1,1,1,4}         92: {1,1,9}
%e A363488     22: {1,5}          58: {1,10}            94: {1,15}
%e A363488     24: {1,1,1,2}      60: {1,1,2,3}         96: {1,1,1,1,1,2}
%e A363488     26: {1,6}          62: {1,11}           100: {1,1,3,3}
%e A363488     28: {1,1,4}        64: {1,1,1,1,1,1}    104: {1,1,1,6}
%e A363488     32: {1,1,1,1,1}    68: {1,1,7}          106: {1,16}
%t A363488 Select[Range[100],EvenQ[#]&&PrimeOmega[#]<=2*FactorInteger[#][[1,2]]&]
%Y A363488 Partitions of this type are counted by A027336.
%Y A363488 The case without high median > 1 is A072978.
%Y A363488 For mode instead of median we have A360015, high A360013.
%Y A363488 Positions of 1's in A363941.
%Y A363488 For mean instead of median we have A363949, high A000079.
%Y A363488 The high version is A364056, positions of 1's in A363942.
%Y A363488 A067538 counts partitions with integer mean, ranks A316413.
%Y A363488 A112798 lists prime indices, length A001222, sum A056239.
%Y A363488 A124943 counts partitions by low median, high A124944.
%Y A363488 A363943 gives low mean of prime indices, triangle A363945.
%Y A363488 Cf. A025065, A215366, A238495, A326567/A326568, A344296, A359889, A359908, A360005, A363486, A363487, A363944.
%K A363488 nonn
%O A363488 1,1
%A A363488 _Gus Wiseman_, Jul 06 2023